First, Gm of the circuit is a function of (W/L) of the input transistor and the tail current. Correct, provided the transistor bias conditions are constant. i.e., your overdrive value is constant. you can refer the graph in razavi text book.
Second, is also correct. Because if you increase the common mode input level, there is no need to change in differential gain. differential gain is the gain we get when a small excitation given at input. common mode voltage is the common voltage at two inputs. both are different and independent.
third, your point is correct. but when u increase Vgs, Vds also changes and it compensates the increase in Vgs to mainitain the constant current in the current equation. But increse in (W/L) is independent of any other parameter, and so gm increases.
hope i cleared ur doubt.